multijimbo

writing long hand in the modern age

•1 May 2016 • 2 Comments

Those of you who know me, know that I carry around a paper journal and that I do a lot of writing long hand. I’ve always done a lot of writing long hand, but I was thinking recently about the current generation of students. What follows will, I suspect, prove to be fodder for further discussions and so I’m going to touch on a lot of things, diving into none of them deeply. I should also note that what comes below, as all the posts in this blog, are my personal views and speculations.

For coursework, the essays and project work that we ask students to do during the semester, we allow them to use word processors or perhaps mathematical typesetting programmes such as LaTeX. In fact, we tend to go a bit further and we require them, by and large, to produce electronic versions rather than write things out by hand. The question is, why?

None of the documents I produce for the consumption of others are written out long hand. This isn’t just because my handwriting tends to be somewhat indecipherable when I’m writing freely. It’s also because I find, as I suspect many others find, that drafting and editing and redrafting is far far easier when using a word processor or suchlike. I find a certain appeal to writing things out by hand, but this is a private appeal.

And so, I have come across a question. Why do we ask students to write out 2 hour examinations long hand? For a few subjects, such as mathematics, I see an argument, because typesetting mathematical symbols can be very time consuming. But for many subjects, we are asking students to do something that they would not ordinarily do, and if fact that we as the teachers might not ordinarily do.

There is a significant logistical answer to this question, namely the difficulty in having enough machines which are sufficiently secure to be appropriate for examination conditions. One issue here is that having a sufficient volume of machines to use during examination time would be difficult to justify, as they wouldn’t be used at other times of the semester or the other times of year. Letting students use their own machines, even under supervised conditions, would create the opportunity for some students having greater access to materials and materiel than others, depending on what files folk would have on their machines and what access they would have to the internet, and it would be impossible to effectively police this.

But this then just leads us back to the question that started us along this particular line. Namely, why do we use 2 hour examinations at the end of the semester to gauge what students have learned over the course of a semester. This is a much more difficult question. On the one hand, having an end of semester examination allows us as teachers to test students under controlled conditions: do these questions, in this time, starting either with a blank examination book or having access to some set collection of material, formula sheets, et cetera.

But on the other hand, this sort of timed examination is not the sort of thing that most people face once they finish their university programme. A few people will, when they take professional examinations, but a relatively few. And so I’m starting to speculate, why do we need end of semester examinations, and if we move to something else, what might that something else be.

I am not the first person to think about this, not by a long way, and the increase of coursework only or coursework majority modules is a significant sign that this is not only something that people are thinking about, but it is something that people are working on. And so let’s leave this one for the moment.

The next question, moving back one more step, is why we structure the information we deliver by the semester. Again, there are strong reasons for doing so, and not just the inertia of history keeping us doing the same thing over time. But I do think that there is an interesting point here for ponderation, namely how we structure the ways in which we structure the material we deliver. And this isn’t even getting into the question of how we deliver material, which is a massive question in its own right.

One thing that is interesting is that the deeper we dig into these questions and the more we start to question some of the fundamental ways in which higher education institutions are structured educationally, there are institutions doing things differently. But I do think that universities with large numbers of students face particular challenges. Over time, we’ll try and come back to some of these, but I think I need to do a bit of reading first, to find out what the current state of the art is.

I suspect that for you, as for me, this has been a somewhat unsatisfactory point at which to conclude. I have raised a lot of questions, with no clear answers, but I do think that the shape of higher education is changing. And I think it’s going to be an exciting ride.

Posted in Uncategorized

why we do mathematics

•1 May 2016 • Leave a Comment

A colleague recently asked, in the course of an unrelated conversation, for my views on why society should fund pure mathematics. Particularly during hard economics times, it’s a reasonable question to ask, why do we as a society fund the things we fund. On the other hand, it is a question about which I have a strong vested interest and a significant bias.

She asked me as I was preparing to talk to a group of pre-university students, giving them reasons that they should consider studying mathematics at university, and the combination of the 2 questions got me thinking.

I am by inclination and training a pure mathematician. I will admit to having dabbled in a few bits of more applied areas of mathematics, but I like geometry and I like studying geometry for its own sake. I enjoy groups and like studying groups for their own sake. I enjoy the abstraction and I love the art of being a mathematician.

My first answer is the same as it would be for most academic endeavours, the subjects covered by the departments that make up universities. It seems bizarre to me that as a society, we would ever be willing to say, we can stop now because we’ve learned enough. It seems bizarre to me that we as a society would ever be willing to say, we have a sufficient understanding of the the inner workings of the physical realm and the realm of ideas, of the human mind and our place in the wider world, that we don’t need to learn any more.

We will never come to a point where we know everything, where we’ve learned enough. Part of this is developing our understanding, and part of this is creating art. There is art in exploring ideas, navigating through possibilities, making connections between hitherto unrelated ideas and concepts.

But beyond this, for mathematics of all stripes and types, there are other reasons.

Mathematicians predict the future. We construct models that allow us to extrapolate from the past and the present, based on the information we have and based on our understanding of the mechanisms of the world, and make reasonable guesses of what the world might be like in days, or weeks, or months. We bear witness to this every day, when we hear the weather report and learn what the future might hold for us.

Mathematicians reconstruct the past. My favorite example here concerns phylogenetic trees. We start with evolution and natural selection. If we make the assumption that life began once on earth, or at least that it is only the descendants of one beginning of life, which I naively believe is the more reasonable assumption, then life is a tree. Where the branches split from one another mark where one species gives birth to two, and so on, and so on.

Viewing things in the present day, we don’t know where are the branches in this tree. But if we know far apart the current species are (say, in how different their DNA is one from another), then we can reconstruct this tree, and in doing so, we can reconstruct the history of all life.

Mathematicians explore the unseen. I like the Radon transform as an example of this. A CAT scan works by shooting beams of radiation through an object with variable density. Some highly sophisticated mathematics allows us to reconstruct the density of the object from the absorption of radiation along each of these beams.

Mathematicians are explorers. Our explorations link the inner world of the mind, with its notions of beauty and structure and order, with the outer world in which we live, allowing us to create models to understand the rhythms of the world. We forge unexpected connections, and even the most abstract and obtuse piece of mathematics has the possibility of shedding light on a poorly understood (or even well understood) corner of our world.

And that I think is sufficient reason.

Posted in mathematics

the language of mastery versus the understanding of the student

•1 May 2016 • 6 Comments

I’ve been thinking about a point I raised in a couple of earlier posts a third meditation on being a teacher and vocabulary and determining the meanings of words There, I started speculating about the difficulties an expert might have in teaching a beginner and how we, how I might get around those difficulties.

Some of the difficulties are relatively straightforward to see. If I have been exploring a topic for an extended period time, mathematical or aikido related or indeed anything else, then how I view that topic changes over time. The aspects of that topic I’m interested in are not the same topics as a beginner might well be interested in. Or indeed should be interested in. And the distance between me and the beginner will grow with time.

So. What do I do about this growing distance. And therein lies the rub. Because however hard it is to see at this moment what to do about this distance, handling this distance will only grow greater with time. After all, I’ll continue to explore and develop my own understanding, and each time I work with a new group of beginners, they’ll be starting at the beginning.

This is not a new question to me. This is shoshin, beginner’s mind, and the question of how to develop the beginner’s mind.

So what I would like to do is to develop a strategy. And this strategy will need to be linked to what I’m doing in a fourth meditation on being a teacher, part 2 in terms of redesigning and reconsidering the structure of the class I’ll be teaching again in the autumn. And yeah, I like spending time contemplating an issue, but I’m not as good at actually coming up with the means by which I can start to tackle the issue.

Again, it’s easy to say that I’d like to develop a strategy. But then we come to the crux of this whole discussion. What is the thing I should do first. And what is then the thing I would do second. And third. What are the steps I will take on the journey of a thousand miles.

This whole train of thought bears on what it means to be a teacher, as well. I’ve come to realize that while part of being a teacher is the structuring of a set of material, a collection of facts and processes for the analysis of those facts, a significant part of being a teacher is charting a straight and narrow path for my students, missing some of the dead ends and cul de sacs that I encountered during my own journey. Not all of them, of course, since lots of learning takes place in these dead ends and cul de sacs.

So here’s a first thing for the next time I teach, be it rolling in aikido or the construction of a proof in mathematics. Break it down into small pieces. Do this, then this, then this, piece by piece, step by step.

But that’s not enough. For me, the pieces have context, but the students haven’t yet developed that context. I understand where the pieces come from, why these pieces are important but other pieces, however obvious they might be, we don’t consider. And this is something I need to explain.

I need to separate the useful bits of this context from the non-useful, misleading bits of this context. And this is perhaps the more important thing to do.

Facts are cheap and this means that the nature of education is changing. (But that’s a separate discussion.) More than structuring the material itself, the facts relevant to the topic of the class, providing this context is where the teacher adds value for the student.

Posted in aikido, mathematics
Tags: aikido, mathematicians, teaching, vocabulary

the necessity of the sith

•9 April 2016 • Leave a Comment

So where to start? After seeing Star Wars episode VII: The Force Awakens, for what will be the first of many times, I was struck by an idea for something to blog about. Start with the joke about the similarities between the Force and duct tape, replace the duct tape with mathematics, and thusly entertain the readers.

But then, procrastination (and see something I’ve been meaning to do for a while for more on that particular topic) and the idea got put on the pile of things awaiting attention. The more I thought about it, the more I began to doubt my original plan. Comparing duct tape and mathematics is only one part of the issue. The other is that I am beginning to doubt that the Force has a light side and a dark side.

Heretical as this might sound, at least to those of you who declared yourselves as Jedi in the previous census, I don’t believe that the Force has a light side and a dark side.

Before going further, though, I should say that I have not done any reading through the now extensive literature of the extended Star Wars universe, nor watched any of the animated films or series. At least, none beyond The Splinter of the Mind’s Eye, which came out before episode V. So perhaps this is something that’s been kicked around therein.

So why no light side and no dark side. First of all, I believe that nature abhors such a dichotomy. A light side and a dark side leaves no space for the ambiguity that pervades our intentions and actions.

But even beyond this, I am not convinced that we are so significant to the Force that it moulds itself to our petty concerns and squabbles over power. I prefer to think that what we view as the light side and the dark side are the reflections of ourselves that we see in the mirror of the Force. We create the light side and we create the dark side, and the Force just abides.

I will admit that part of the reason I prefer this interpretation is that this makes our relationship to the Force into an active relationship. We are not passive tools of some dichotomy that exists beyond us. Instead, we make a choice of the direction towards which we tilt. It also allows for the possibility of a dynamically unstable middle ground, a grey servant of the Force.

The necessity of the Sith then is that we each have something of a dark side, the thoughts and desires that inform the entertainments with which we like to spend our time. The necessity of the Sith is that we are human, or one of the other remarkably similar races that fill the ranks of both the Jedi and the Sith.

Posted in Uncategorized

a job interview versus a martial arts grading

•30 March 2016 • 1 Comment

This is something I’ve been pondering for a while now, and it’s something that first came to my mind about a year ago, when I was preparing for a promotion interview. I’ve only had a few interviews in the course of my career, but I’ve also had 8 aikido gradings as well, and I was struck by an interesting difference between the two sorts of events.

An interview is a bottleneck situation. Several people are competing for a single job. The panel that is interviewing takes the documentary evidence of curriculum vitae, perhaps with other statements depending on the job, perhaps with a presentation, and after talking to each of the candidates, makes a decision.

An aikido grading is a very different sort of situation. In an aikido grading, it is possible that everyone who is grading passes, and it is possible that everyone who is grading fails the grading. Most times, it falls somewhere in between. In a grading situation, people have spent significant time and effort in their preparation, and again the grading panel makes their decision.

The significant difference between the two is the purpose. An interview is to choose one from many and often, many or all of the candidates will be suitable for the job. A grading on the other hand is to judge individuals against a set of criteria, not against one another. Even so, I suspect there are conditions in which one candidate in a grading performs so well (or so poorly) as to shade the performance of the others.

The reason that all of this came to mind is that while I was preparing for my promotion interview, I came to the realization that this promotion interview was a grading, rather than an interview as described above. There were a number of us who were going through the promotion interview process, but there was no limit on how many could be promoted. Rather, like a grading, we were individually and independently being judged against a set of criteria.

And this then led me to something else. Why put myself through the grading process, either the promotion interview process or the aikido grading process. After all, the process of preparation for these gradings is somewhat stressful.

It may be different for others, but for me, there was one common reason that linked the promotion interview and the aikido gradings. For me, it is helpful to know, it is helpful to have the feedback, that in the eyes of my senior and more experienced colleagues, both at the University and in aikido, I am doing what is expected. And as I become one of those more experienced colleagues to others, I will bear this distinction in mind.

Posted in aikido, mathematics
Tags: grading aikido promotion interview

a second meditation on supervillainry

•28 February 2016 • 3 Comments

As I work through my backblog, I come across things I started some long time ago and meant to finish long before now. This is one of those. Supervillainry, the bane of James Bond and Batman, mocked by many, and yes, somehow my memories of supervillains always come back to Wile E Coyote, super genius, ever incapable of catching the Road Runner.

And so, the question now becomes, what to do. What actually to do. Let’s suppose that I were so dissatisfied with my job that I was considering the path of supervillainry as a change of career. An actual change of career. Something different to do tomorrow, to wake up to tomorrow. I am aware of the old Taoist saying, the journey of a thousand miles begins with but a single step. But what of the second step? The hundredth? The thousandth or millionth?

As far as I am aware, there is no SPECTRE in the modern world. There is no secret organisation, no arch villain, no evil super genius planning on world domination. There is nowhere for me to apply for an entry level position.

What are my options? The most probable option is to do nothing. Live my life, stay on the path I currently walk along, and forget this dream of such a drastic career change. Makes for a boring story, that. The main character ignores the impetus to change and decides to go into the office the next day, same as always. Not much of a readership, I expect. Not much of a following for that line of inaction.

Or I could take the drastic decision to set up such an organisation. So what if I were to do this. Every supervillain organisation needs to have a mission, a calling, the basic principle that underlies everything.

Though I have a fondness for doomsday devices, I don’t think they’re appropriate for the task at hand. SPECTRE was driven by a love of money and power, particularly power, but I have no great desire for power. Rather, I have a desire for change. I would like the world to be different. But which of the so many ways to be different do I want to choose?

So let’s combine 2 things. This idea will no doubt mutate over time, as I back myself into logical corners and need to backtrack back to the safety of something that might work. How to combine supervillainry and mathematics, that’s the challenge I set for myself.

On the very fanciful end, there is the supervillain with his Banach-Tarski machine, an idea I suspect that someone has used before now. Or perhaps something more sedate, descending into some labyrinth of the abstract. So is there a way to combine supervillainry and mathematics? Let’s see what we can find.

Posted in mathematics, writing ephemera
Tags: mathematicians, supervillain

a fourth meditation on being a teacher, part 2

•21 February 2016 • 3 Comments

And so, we come back to spend a bit of quality time with my grand plan of fundamentally redesigning my teaching. I think that the place to start is by looking deeply at the things I’m currently doing. From time to time, it behooves us all to lean back, close our eyes, and examine our fundamental assumptions, those bits of bedrock on which we base everything we do.

As mathematicians have done for centuries, I lecture. But what does this mean in practice? It starts with all of the things that go into preparing for the act of lecturing. I spend time in my office, or on the train, or wherever I happen to be, structuring the material. I work through examples, some standard, some non-standard, some weird and strange to illustrate a particular point I wish to make, so that I can present them smoothly during the lecture itself. I construct exercises to allow the students to engage with the material.

But what of the act of lecturing itself? I am beginning to think that in some ways and at some times, lecturing is the wrong description, and that transcriber might be more appropriate.

Reading and writing are very recent acquired talents in human history, both only a few thousand years old, and both only much more recently expected of members of the general populace. Speaking and listening, though, are much older, and so a question worth asking is, do we learn better when listening or when reading? I don’t know the answer, but if you do or can point me in a direction, I’d be interested in getting a reference for who’s explored this.

And this is relevant for the discussion we’re having. If it turns out that the answer to my question above is no, there is no difference in learning between listening and reading, then this begins to undercut the reason for having a standard lecture, where the lecturer talks and the students listen. Until I learn otherwise, though, I choose to believe the answer is yes.

This is not to say that there is no place for a lecturer and students being in the same room at the same time. Only that in order for the lecture to add some value to the student experience, there would need to be something else. There would need to be interaction between lecturer and student.

And so there is where I am at the moment. What are the things that I can do to add value to that time when I’m in the room with my students. And I’ll go back to the beginning, to the choice of the material I talk through versus the material I make available; to how I present this material; to how I encourage my students to be active participants versus passive recipients.

Looking back, I’ve been experimenting with things for years now, but not in a particularly coordinated way. And so, before I next teach, I think that quest will be to impose some coordination on my activities in and outside of the class room.

Posted in mathematics

habits, the kudzu of the brain

•7 February 2016 • 2 Comments

Growing up in Georgia, a common site on our drives down to the coast to visit my grandparents was a field covered in kudzu, often with a suspicious indistinct bump in the middle where we suspected a house still stood, covered when it wasn’t looking.

I had the intention of writing a post herein starting from that image, of the irresistible and irrepressible progress of kudzu across a landscape, consuming everything in its path, a terrifying invasive species that was consuming the South.

But then, I did some reading and I discovered that much of what I knew about kudzu was rumor and speculation, and in fact I didn’t know much at all. For those who are interested in the side of right in the battle between myth and fact, I recommend Bill Finch’s recent article The True Story of Kudzu, the Vine That Never Truly Ate the South

As much as I enjoyed finding out the truth about kudzu, it did rather puncture the analogy that I’ve constructed in my head, about habits spreading in my mind the same way that kudzu spreads across the southern landscape.

I struggle with some of my habits. I struggle with procrastination, though I’m working on it and I’ve discussed this elsewhere. And I struggle with the development of new habits, to which my erratic posting history for this blog will strongly attest.

So perhaps my analogy was fatally flawed from the get go. Perhaps the view I expressed in the title of this short piece is wrong and I need to adopt a different view of habits and my relationship with my habits. Food for thought.

Posted in bizarre and random things
Tags: habits, kudzu, procrastination

vocabulary and determining the meanings of words

•7 February 2016 • 2 Comments

We had a lovely aikido class this morning, and an issue that has been kicking around in my head for quite some time crystalized at one point in our discussions of one of the techniques on which we were working.

For quite a long time now, I’ve pondered the issue of how an experienced person should teach something to a beginner. This relates closely to issues around teaching, but that’s another thread and I want to speculate on one specific aspect of this.

Let me start by saying that I do believe that it’s possible for an experienced practitioner can teach a beginner. But what I am coming to see, and what crystalized for me this morning, is the importance of developing vocabulary.

As we spent time learning something, as we spend time journeying from beginner to experienced practitioner, we develop a vocabulary in common with those with whom we’re undertaking that journey.

This has long been clear to me as a mathematician, if only because as mathematicians we have developed a structure in which to discuss mathematics. We set our axioms and our definitions. From these axioms and definitions, we prove lemmas and propositions and theorems. We build grand edifices, elaborate structures of logic.

But it is hard for the novice to explore these elaborate structures on their own. It is hard for the beginner to understand the subtleties being explored because of the sheer elaborateness of the structure within which we’re working.

The same occurred to me this morning, during our aikido class. We are working to understand how we are built underneath our own skins, muscles and tendons, bones and ligaments. And we are working to use this understanding to improve our aikido, how we move and how we move our uke through our own movement.

Again, we use words and phrases to try and describe what we are doing: relax the shoulders is one of the most common, and it is only now that I am beginning to understand what we mean by it. I am also sure that my understanding, while greater than a beginner’s understand, is as a beginner’s understanding to those who are my seniors.

And that fact, that I will always have so much to learn, that there will always be vistas to explore beyond those I’ve already explored, is a large part of why I find these undertakings fascinating.

Posted in aikido, bizarre and random things, mathematics
Tags: aikido, experience, mathematicians, vocabulary

approximation and estimation in the digital age

•31 January 2016 • Leave a Comment

This is a wildly speculative post. It may be that some of the speculations below have been well researched by others, and if they have been and if you would happen to know, I would be interested in knowing who those others might be and what they’ve written on these.

There are things I believe to be true, facts about the world as it currently is, facts that I understand on an intellectual, rational level. When I think about these facts and hold these thoughts for a few seconds in my head, I say to myself, yes I see how that is true. They make sense to me.

One of the most basic of these things I believe to be true is that the power of the written word is diminishing. And before you get angry and start ranting, let me ask for your patience and for a bit of time to explain what I mean.

The human activity of writing is a few thousand years old. Before the invention of alphabets and writing, and probably even for some short time after, we humans carried our knowledge around in our heads, in the stories that we told one another around our campfires.

Perhaps, as some suspect, we used the stories we told to encode facts we have laboriously and painstakingly discovered about our world. It was reading Hamlet’s Mill by de Santillana and von Dechend that first raised this possibility in my mind, and this idea of how humans use myth rang true with me. But regardless, without writing we would have had no choice but to keep and transmit our history and our acquired knowledge in our stories.

This changed with the invention of the written word. Much to the resentment of some of our predecessors, it became possible to store our history outside of our heads. Facts about our world were still difficult and expensive to acquire, but the facts so acquired could then be safely stored, subject only to the occasional transcription error or catastrophic all consuming fire.

Universities and monasteries became the repositories of human knowledge, holding our accumulated facts. This repository nature of universities persists and this persistence has some implications for universities, but this is something to explore elsewhere.

My generation grew up reading. We had television as well, and movies, but both are relatively static media, in the sense that production costs are high and the product is not particularly interactive. When we needed to find a fact, we went to the library, or to an encyclopaedia at home, and we read the fact and around that fact.

One of the joys of my early days was doing exactly this, finding the book on the shelf for which I was searching, and then looking at and looking through the books whose only attraction was that they were near it on the shelf.

But I’m beginning to think that the younger generation doesn’t have this opportunity. There is a disadvantage to using Google, to the current digital age. This disadvantage is that the digital world does not easily lend itself to approximation, and to estimation. With the digital, we come to expect exactness, and exactness is limited.

Perhaps I’m wrong, but one thing that I’ve seen in the students I teach is that they don’t understand how to estimate or approximate. They understand exactness, but they become uncomfortable if taken a bit out from within the lines. I would be interested to know if this is something that others have encountered.

Posted in bizarre and random things